Functional limit theorems for linear processes with tapered innovations and filters

In the paper we consider the partial sum process ${\sum }_{k=1}^{[nt]}{X}_k^{(n)}$, where $\{X_k^{(n)}={\sum }_{j=0}^{\mathrm{\infty }}{a}_j^{(n)}{\xi }_{k-j}(b(n)),k\in \text{bz}\},n\ge 1,$ is a series of linear processes with tapered filter $a_j^{(n)}=a_j\text{ind}[0\le j\le \text{ł}(n)]$ and heavy-tailed tapered innovations ${\xi }_j(b(n),j\in \text{bz}$. Both tapering parameters $b(n)$ and $\text{ł}(n)$ grow to $\mathrm{\infty }$ as $n\to \mathrm{\infty }$. The limit behavior of the partial sum process (in the sense of convergence of finite dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with non-tapered filter $a_i,i\ge 0$ and non-tapered innovations. We consider the cases where $b(n)$ grows relatively slow (soft tapering) and rapidly (hard tapering), and all three cases of growth of $\text{ł}(n)$ (strong, weak, and moderate tapering).